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DEVELOPMENT OF FINITE-DIFFERENCE TIME-DOMAIN (FDTD) /
PERIODIC BOUNDARY CONDITION (PBC) SOFTWARE FOR MODELING MICROWAVE ABSORBER MATERIAL
AT X-BAND FREQUENCY
MUHAMMAD AFIQ BIN MD SALIMIN
This Report is Submitted in Partial Fulfillment of Requirement For The Bachelor Degree Of Electronic Engineering (Telecommunication Electronic)
With Honours
Faculty of Electronic and Computer Engineering Universiti Teknikal Malaysia Melaka
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“I hereby declare that this report is the result of my own work except for quotes as cited in the reference”
Signature : ………
Author : MUHAMMAD AFIQ BIN MD SALIMIN
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“I hereby declare that I have read this report and in my opinion this report is sufficient in terms of the scope and quality for the award of Bachelor of Electronic
Engineering (Telecommunication Electronics) with Honours”
Signature : ………
Author : MR FAUZI BIN MOHD JOHAR
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ACKNOWLEDGEMENT
In the name of Allah, the Most Gracious and the Most Merciful Alhamdulillah, all praises to Allah S.W.T for His blessing which able me to complete this final year project. First of all I would like to address my special appreciation to my supervisor, Mr Fauzi Bin Mohd Johar for all the guidance gave to me from the start until the end of time for me in completing this final year project. Also thank you for all the support and advice given when teaching me in completing this thesis writing
I also want to thank to my family for the moral support given to me for the whole time of my study. The full support had help to raise my passion and spirit to finish all my work perfectly especially in writing this thesis. Besides that, I would like to express my gratitude to all my friends especially my classmates that had gave me ideas during the time I am doing this project. All the opinion, suggestion and helps given will never be forgotten with all the memories left.
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ABSTRACT
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ABSTRAK
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TABLE OF CONTENT
CHAPTER TITLE PAGE
PROJECT TITLE i
VALIDATION REPORT STATUS FORM ii
DECLARATION iii
SUPERVISOR COMFIRMATION iv
ACKNOWLEDGEMENT v
ABSTRACT vi
ABSTRAK vii
TABLE OF CONTENT viii
LIST OF CHARTS xi
LIST OF FIGURES xii
LIST OF ABBREVIATIONS xiv
LIST OF APPENDICES xv
I INTRODUCTION
1.1 Project Overview 1
1.2 Problem Statement 2
1.3 Objective 4
1.4 Scope 5
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II LITERATURE REVIEW
2.1 Finite-Difference Time-Domain (FDTD) 7 2.1.1 The Finite Difference Time Domain (FDTD)
Basic Equation 8
2.1.2 Yee Cell’s Concept 8
2.1.3 Discretization of Finite Difference Time
Domain (FDTD) Equation 11
2.2 Absorbing Boundary Condition (ABCs) 17 2.2.1 Convolution Perfectly Matched Layer (CPML) 18
2.2.1.1 Convolutional Perfectly Matched Layer Regions and Associated Field Updates
for a Three-Dimensional Domain 18 2.2.2 Perfect Electric Conductor (PEC) 27 2.3 Periodic Boundary Condition (PBC) 28 2.4 Frequency Selective Surface (FSS) 30
2.5 Radar Absorber Material (RAM) 31
III METHODOLOGY
3.1 Information Acquisition Methods 34
3.1.1 Books 34
3.2.1 Articles, Journals and Reports 35
3.2 Flow Chart of Project Planning 35
3.3 Flow Chart of FDTD Program 37
IV RESULT, ANALYSIS AND DISCUSSION
4.1 Frequency Selective Surface (FSS) Analysis 40
4.1.1 Changing the Patch Shape 41
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4.1.1.2 Square Loop FSS 43
4.1.1.3 Jerusalem FSS 44
4.1.1.4 Fixed Shaped Frequency Selective
Surface (FSS) Discussion 45
4.1.2 Number of Iteration Analysis 46 4.1.3 Value of Dielectric Constant Analysis 47
4.1.4 Random Shape Analysis 48
4.1.4.1 Review of Random Shape 48 4.1.4.2 Review of GRM Function 49
4.1.4.3 Random Shaped FSS 50
4.2 Radar Absorber Material (RAM) Analysis 51
4.2.1 Fixed Shape Analysis 52
4.2.2 Random Shape Analysis 54
VI CONCLUSION AND RECOMMENDATION
5.1 Conclusion 57
5.2 Recommendations 58
REFERENCES 60
APPENDICES
Appendix A 62
Appendix B 63
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LIST OF CHARTS
NO TITLE PAGE
3.1 Flow Chart of Project 36
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LIST OF FIGURES
NO TITLE PAGE
1.1 Project Structure 6
2.1 3D FDTD Computational Space Composed of Yee’s Cell 9 2.2 Arrangement of Field Component on Yee’s Cell 9
2.3 Electric Field Component on Yee Cell 11
2.4 Magnetic Field Component on Yee Cell 11
2.5 PML Component Region 17
2.6 Periodicity of Periodic Structure 28
2.7 Double Square FSS 31
2.8 Cross Dipole FSS 31
2.9 Stealth Plane 32
2.10 Anechoic Chamber Room 32
2.11 Radar Absorber Material Structure 33
4.1 Cross Dipole FSS in Fortran 42
4.2 Cross Dipole FSS in CST 42
4.3 Graph of Transmission Coefficient for Cross Dipole FSS 42
4.4 Square loop FSS in Fortran 43
4.5 Square loop FSS in Fortran 43
4.6 Graph of Transmission Coefficient for Square Loop FSS 43
4.7 Jerusalem FSS in Fortran 44
4.8 Jerusalem FSS in Fortran 44
4.9 Graph of Transmission Coefficient for Jerusalem FSS 44
4.10 Graph of Number of Iteration Analysis 46
4.11 Graph of Changing Dielectric Constant 47
4.12 Example of Random Shape Patch 48
4.13 Patch With Critical Points 49
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4.15 Random Shape for FSS Analysis 50
4.16 Graph of Transmission Coefficient for Random Shape FSS 51
4.17 Jerusalem RAM in Fortran 3D View 52
4.18 Jerusalem RAM in Fortran 2D View 52
4.19 Graph of Reflection Coefficient for Jerusalem Shaped RAM 53
4.20 Random Shaped RAM in 3D 54
4.21 Random Shaped RAM in 2D 55
4.22 Graph of Reflection Coefficient for random Shaped RAM 56
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LIST OF ABBREVIATIONS
FDTD - Finite Difference Time Domain
PBC - Periodic Boundary Condition
FSS - Frequency Selective Surface
RAM - Radar Absorber Material
PEC - Perfect Electrical Conductor
ABC - Absorbing Boundary Condition
PML - Perfectly Matched Layer
CPML - Convolutional Perfectly Matched Layer
CST - Computer Simulation Technology
GRM - Geometry Refinement Method
FFT - Fast Fourier Transform
CEM - Computational Electromagnetic
GA - Genetic Algorithm
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LIST OF APPENDIX
NO TITLE PAGE
A Gantt Chart 62
B Product Poster 63
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CHAPTER I
INTRODUCTION
This chapter will give brief explanation about this project. The overview in this chapter will explain overall concept of the project. Then objective is stated to show what is the project aim related to the problem statement explained. Scope part will explain what is covered and uncovered topic in this project.
1.1 Project Overview
This project will introduce about the Finite Difference Time Domain (FDTD) method that used to solve Maxwell’s equation. The fundamental and application of the FDTD carries the numerical calculation of the Maxwell’s equation in Cartesian coordinate system. These equations are used to develop Finite Difference Time Domain (FDTD) algorithm for modeling Radar Absorbing Material (RAM). Before implementation of Maxwell’s equation into Finite Difference Time Domain (FDTD) algorithm, the equations were discretized to make sure that it is compatible to be use for modeling a full wave electromagnetic structure.
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Using the combination of FDTD and PBC, only one unit cell need to be model and this can save more memory space of the computer compare to simulating the whole project that includes so many unit cells. Besides, the computational time taken will be more efficient when this method is done to simulate several periodic structures. The Frequency Selective Surface (FSS) that implemented on the Radar Absorbing Material (RAM) also can be considered as periodic structure since every unit cell has identical patch. The identical unit cell is repeating regularly for certain dimension length of the FSS structure.
Since the RAM structure implement the FSS concept for patch in front of it, the FSS will be analyzed first to see the effect of response when several parameters are changed. Then, with the guided of result obtained in FSS analysis, the RAM is designed and again the analysis is carried out to see how the value of parameters will affect the response of the structure to the electromagnetic wave that applied.
The FDTD iteration will obtain reflected wave values in voltage and ampere unit. Then the scattering parameter, which is the reflection coefficient of the wave is calculated after the FDTD iteration is finished with aid of Fast Fourier Transform (FFT). Scattering parameter (S1,1) is then plotted to see the response of the Radar Absorbing Material. The result obtained is finally compared to the result given by commercial software to see the accuracy.
1.2 Problem Statement
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In existence of computer, there are several powerful technique and method to be chosen from to perform evaluation, analyzing and designing analysis. This includes analysis that involves so many numerical calculations in determining the result such as designing the electromagnetic devices or structures. Since it is hard to do manually by experimental method, so modeling in computer software is the best method to be chosen.
The implementation of Computational Electromagnetic (CEM) method for the analysis will overcome the disadvantages of experimental method by reducing the test cost and it also versatile and accurate. This means that modeling in computer has many advantages. The CEM method consists of integral and differential equation in time domain. The example integral equation is Method of Moment (MoM) and for differential is Finite-Difference Time-Domain (FDTD).
Currently, FDTD has gained tremendous popularity as a tool in solving Maxwell’s Equation. Therefore some people developed software with function to solve Maxwell’s equation problem. This software is then commercialized to be used by public and the developer may gain much profit on selling their commercial software. But, there is still some disadvantages exists in the commercial software.
The commercial software has limitation in its function. In term of solving Radar Absorbing Material (RAM) problem, the shape of the Frequency Selective Surface cannot be made randomly. User still can do whatever shape that they want. But, complex shape is very hard to be created which makes the designing process is very hard. Therefore it is harder when optimization process need to be carried out. A lot of time is consumed just to get the desired result of the project.
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The commercial software sold by developer is normally too expensive for normal people. Some people will find easy way by using pirate license to activate the software. But this is unethical action and the people that practice this might be sued by the developer. Some commercial software will give free trial period for the user to use and the software can be downloaded from the internet. But it only can be use in a limited time with limited usage of function inside the software. After the trial period end, user will be unable to use the software anymore.
For a student, it might be impossible for them to buy this expensive commercial software. Sometime they might have to go to laboratory which provides licensed commercial software for them to use. If their project takes very long time to finish, this can be a problem to them since they have to go to laboratory every time they want to do the project.
1.3 Objective
There is one objective for this project which is:
1. To develop 3D Finite Difference Time Domain (FDTD)/Periodic Boundary Condition (PBC) program for modeling Radar Absorbing Material using Fortran software in X-Band Frequency
The main objective in this project is to develop FDTD/PBC program. Therefore the concept of 3D Finite Difference Time Domain (FDTD) method which uses Maxwell’s equation in the algorithm must be studied for understanding. Then using the FDTD, Radar Absorbing Material is modeled by developing program in Fortran software. The program is able to analyze the wave absorbed by the Radar Absorbing Material and the reflection coefficient is obtained at the end of the simulation. Lastly the scattering parameter of the reflection coefficient obtained is compared to the result given by commercial software.
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1.4 Scope
There are some main theories and concepts that is taking under consideration in this project. Firstly the project is majorly applying 3D Finite Difference Time Domain (FDTD) concept to solve two types of Maxwell’s equation which are Ampere Maxwell law and Faraday Maxwell law. The three dimension perspective is taken so that the modeling will produce real shape object which cannot be done in two dimensional view.
The two equations are derived in term of Cartesian coordinate until come to the discretization of the equation. By implementing Yee Cell’s concept, the discretization will be easier since the position of electric field and magnetic field can be easily recognized on the Yee Cell. Electric field is on the edge of the Yee Cell and magnetic field is at the middle of the Yee Cell. The Yee Cell concept is the best to be use together with FDTD method.
Then the modeling is for Radar Absorbing Material (RAM), not for antenna or other application. Frequency Selective Surface is implemented together with the Radar Absorbing Material. Good understanding about these two structure is needed so that the theoretical knowledge will be easily applied into the coding to build the program. In modeling the Radar Absorbing Material, Fortran language in used instead of Matlab, C++ or C language. This is because Fortran language will produce execution file which is not available on other language. Besides, Fortran language is easier to be understand.
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The other concept that is used in this project is the absorbing boundary condition. The type of boundary used is Convolution Perfect Matched Layer (CPML) instead of Perfect Matched Layer (PML). CPML has simpler equation and easier to use. Although the equation of CPML is based on PML, but this project only focusing on the CPML equation and does not cover any PML equation. Only the concept of PML is taken since it is same for CPML and the function also same, to absorb electromagnetic waves until it fully attenuate.
1.5 Project structure
The project is build up from many components. It can be illustrates as in the Figure 1.1 below
Figure 1.1: Project structure
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CHAPTER II
LITERATURE REVIEW
This chapter will describe the fundamental concept and theory of the FDTD methods in solving Maxwell’s curl equation in time domain. The derivations of the equations, which called as discretization are also shown for the electric and magnetic field. The Absorbing Boundary Condition (ABC) and Frequency Selective Surface (FSS) for Radar Absorbing Material (RAM) also will be explained. Finally, the Periodic Boundary Condition (PBC) and Perfect Electric Conductor (PEC) also summarized related to this project.
2.1 Finite Difference Time Domain (FDTD)
Finite Difference Time Domain (FDTD) is one of the methods for computational electromagnetic (CEM) applications. FDTD is a tool for solving Maxwell’s equations and the calculations made are in time domain. Therefore FDTD is very suitable to solve the differential time domain Maxwell’s equation which used in this project. FDTD is also related to numerical analysis that implements numerical calculation method in solving problems.
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2.1.1 The Finite Difference Time Domain (FDTD) Basic Equation
To develop FDTD algorithm, first we must understand its equation. FDTD is using Maxwell’s time-domain equation as the starting point. There are two basic equations that we use in here which are Ampere Maxwell’s equation and Faraday Maxwell’s equation [1]-[5].
Ampere Maxwell’s Law
�
� =∇×� −
Faraday Maxwell’s Law
�
� = ∇× −
Where the symbols are defined as: D = electric flux density (C/m2) H = magnetic field (A/m)
J = electric current density (A/m2) B = magnetic field density (V/m2) E = electric field (V/m)
M = magnetic current density (V/m2)
2.1.2 Yee Cell’s Concept
In FDTD method, the problem space divided into small grid that called Yee cells that form a cube like segment. This technique that employs the second-order central difference formula that represented in discrete form of time and space. By applying this technique, the electric and magnetic fields can be solved in a leapfrog manner. It means that each of electric and magnetic field dependent on the neighbor field on each of time steps [1]-[5].
2.1.1 (a)
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Figure 2.1: 3D FDTD Computational Space Composed of Yee’s Cell From the Figure 2.1, it shows how the cell grid composed with the Nx, Ny, and Nz represent the maximum number of cells in the problem space. In designing the object geometry, the space resolution of the object set by the size of the unit cell and the material parameters including permittivity , permeability, electric and magnetic conductivity must be set to distinguish between object and free space.
Figure 2.2: Arrangement of Field Component on Yee’s Cell
